L_0 Isotonic Regression With Secondary Objectives

06/01/2021

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by   Quentin F. Stout, et al.

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We provide algorithms for isotonic regression minimizing L_0 error (Hamming distance). This is also known as monotonic relabeling, and is applicable when labels have a linear ordering but not necessarily a metric. There may be exponentially many optimal relabelings, so we look at secondary criteria to determine which are best. For arbitrary ordinal labels the criterion is maximizing the number of labels which are only changed to an adjacent label (and recursively apply this). For real-valued labels we minimize the L_p error. For linearly ordered sets we also give algorithms which minimize the sum of the L_p and weighted L_0 errors, a form of penalized (regularized) regression. We also examine L_0 isotonic regression on multidimensional coordinate-wise orderings. Previous algorithms took Θ(n^3) time, but we reduce this to o(n^3/2).